Oblate ellipsoid

Surface shape factors are much more difficult to measure than volume shape factors and they are subject to greater uncertainty. One method is as follows. A few individual crystals are observed through a low-power microscope fitted with a calibrated eyepiece, and sufficient measurements taken to allow a sketch to be made of a representative geometric shape, e.g. a parallelepipedon, ellipsoid, oblate spheroid, etc. The surface area of the representative solid body may then be calculated. It should be appreciated, of course, that the result of such a calculation will be prone to significant error. 

The ellipsoid of revolution is swept out by rotating an ellipse along its major or minor axis. When the major axis is the axis of rotation, the resulting rodlike figure is said to be prolate when the minor axis is the axis of rotation, the disklike figure is said to be oblate. 

We designate the length of the ellipsoid along the axis of rotation as 2a and the equatorial diameter as 2b to define the axial ratio a/b which characterizes the ellipticity of the particle. By this definition, a/b > 1 corresponds to prolate ellipsoids, and a/b < 1 to oblate ellipsoids. 

Based on these ideas, the intrinsic viscosity (in 0 concentration units) has been evaluated for ellipsoids of revolution. Figure 9.3 shows [77] versus a/b for oblate and prolate ellipsoids according to the Simha theory. Note that the intrinsic viscosity of serum albumin from Example 9.1-3.7(1.34) = 4.96 in volume fraction units-is also consistent with, say, a nonsolvated oblate ellipsoid of axial ratio about 5. 

Figure 9.3 Intrinsic viscosity according to the Simha theory in terms of the axial ratio for prolate and oblate ellipsoids of revolution.

At 37°C the viscosity of water is about 0.69 X 10"3 kg m" sec" the difference between this figure and the viscosity of blood is due to the dissolved solutes in the serum and the suspended cells in the blood. The latter are roughly oblate ellipsoids of revolution in shape. 

Cylindrical disk R radius of disk (approximation for oblate ellipsoids for which a/b 1) ... 

Fig. 3a,b. The phase behaviour of a system of hard ellipsoids, both prolate and oblate, as a function of the ellipticity, alb, plotted against a the packing fraction, p b the scaled number density, p ... 

A system of particles interacting in this way was studied using a microcanonical ensemble at a scaled density of 3.0 which is close to the transitional density for hard oblate ellipsoids with the same ellipticity (see Fig. 3). At a scaled density of 3.0 the system is found to exhibit isotropic. 

The quantity riV/RT is equal to six times the rotational period. The rotational relaxation time, p, should he shorter than the fluorescence lifetime, t, for these equations to apply. It is possible to perform calculations for nonspherical molecules such as prolate and oblate ellipsoids of revolution, but in such cases, there are different rotational rates about the different principal axes. 

Sanchez, C. Schmitt, C. Kolodziejczyk, E. Lapp, A. Gaillard, C. Renard, D. (2008). The Acacia Gum Arabinogalactan Fraction Is a Thin Oblate Ellipsoid A New Model Based on Small-Angle Neutron Scattering and Ab Initio Calculation. Biophysical Journal, Vol. 94, No. 2, (January 2008), pp.629-639, ISSN 0006-3495. 

Uniaxial deformations give prolate (needle-shaped) ellipsoids, and biaxial deformations give oblate (disc-shaped) ellipsoids [220,221], Prolate particles can be thought of as a conceptual bridge between the roughly spherical particles used to reinforce elastomers and the long fibers frequently used for this purpose in thermoplastics and thermosets. Similarly, oblate particles can be considered as analogues of the much-studied clay platelets used to reinforce a variety of materials [70-73], but with dimensions that are controllable. In the case of non-spherical particles, their orientations are also of considerable importance. One interest here is the anisotropic reinforcements such particles provide, and there have been simulations to better understand the mechanical properties of such composites [86,222],... 

Ellipsoid If the base of a vessel is one-half of an oblate spheroid (the cross section fitting to a cylinder is a circle with radius of D/2 and the minor axis is smaller), then use the formulas for one-half of an oblate spheroid. 

A droplet is initially flattened to an oblate, lenticular ellipsoid and then may be converted into a torus, depending on the magnitude of the internal forces causing the deformation. The torus subsequently becomes stretched and splits into smaller droplets. 

A modified version of the TAB model, called dynamic drop breakup (DDB) model, has been used by Ibrahim et aU556l to study droplet distortion and breakup. The DDB model is based on the dynamics of the motion of the center of a half-drop mass. In the DDB model, a liquid droplet is assumed to be deformed by extensional flow from an initial spherical shape to an oblate spheroid of an ellipsoidal cross section. Mass conservation constraints are enforced as the droplet distorts. The model predictions agree well with the experimental results of Krzeczkowski. 311 ... 

For all three types of dendrimers described above, a flattened, disk-like conformation was observed for the higher generations. However, the molecular shape at the air-water interface is also intimately associated with the polarity, and hence the type of dendrimer used. In case of the polypropylene imine) and PAMAM dendrimers the hydrophilic cores interact with the sub-phase and hence these dendrimers assume an oblate shape for all generations. The poly(benzyl ether) dendrimers, on the other hand, are hydrophobic and want to minimize contact with the water surface. This property results in a conformational shape change from ellipsoidal, for the lower generations, to oblate for the higher generations [46]. 

In the particular case of prolate and oblate ellipsoids, the number of exponentials is reduced to three because two of the three axes are equivalent. The rotation diffusion coefficients around the axis of symmetry and the equatorial axis are denoted Di and D2, respectively. The emission anisotropy can then be written as... 

Patel et al. showed that a Bruggeman exponent of 1.5 is often not valid for real separator materials, which do not have uniform spherical shape.Porous networks based on other morphologies such as oblate (disk-type) ellipsoids or lameller increase the tortuous path for ionic conductivity and result either in a significant increase of the exponent a, or in a complete deviation from the power law. They showed that spherical or slightly prolate ellipsoidal pores should be preferred for separators, as they lead to higher ionic conductivity separators. 

For an oblate ellipsoidal drop with the major axis Dh in a horizontal... 

If a drop of low viscosity moves through a field of corn syrup of viscosity of about 300 c.p., the series of shape changes shown in Fig. 10b will occur. The succession is spherical, ovate, spherical, symmetrical oblate ellipsoidal, nonsymmetrical ellipsoidal and, finally, inverted mush-room-like shapes with an indented rear surface (FI). 

Equation (3-33) shows how the inertia term changes the pressure distribution at the surface of a rigid particle. The same general conclusion applies for fluid spheres, so that the normal stress boundary condition, Eq. (3-6), is no longer satisfied. As a result, increasing Re causes a fluid particle to distort towards an oblate ellipsoidal shape (Tl). The onset of deformation of fluid particles is discussed in Chapter 7. 

The mechanism of mass transfer to the external flow is essentially the same as for spheres in Chapter 5. Figure 6.8 shows numerically computed streamlines and concentration contours with Sc = 0.7 for axisymmetric flow past an oblate spheroid (E = 0.2) and a prolate spheroid (E = 5) at Re = 100. Local Sherwood numbers are shown for these conditions in Figs. 6.9 and 6.10. Figure 6.9 shows that the minimum transfer rate occurs aft of separation as for a sphere. Transfer rates are highest at the edge of the oblate ellipsoid and at the front stagnation point of the prolate ellipsoid. 

The conditions under which fluid particles adopt an ellipsoidal shape are outlined in Chapter 2 (see Fig. 2.5). In most systems, bubbles and drops in the intermediate size range d typically between 1 and 15 mm) lie in this regime. However, bubbles and drops in systems of high Morton number are never ellipsoidal. Ellipsoidal fluid particles can often be approximated as oblate spheroids with vertical axes of symmetry, but this approximation is not always reliable. Bubbles and drops in this regime often lack fore-and-aft symmetry, and show shape oscillations. 

Transfer from large bubbles and drops may be estimated by assuming that the front surface is a segment of a sphere with the surrounding fluid in potential flow. Although bubbles are oblate ellipsoidal for Re < 40, less error should result from assumption of a spherical shape than from the assumption of potential flow. 

The biggest difference between biological particles and ceramic particles in the application of Eq. (4.20) is that while most ceramic particles are spherical ( Ch = 2.5), most biological particles can be modeled as either prolate ellipsoids or oblate spheroids (or ellipsoids). Ellipsoids are characterized according to their shape factor, ajb, for which a and b are the dimensions of the semimajor and semiminor axes, respectively (see Eigure 4.17). In a prolate ellipsoid, a > b, whereas in an oblate ellipsoid, b > a.ln the extremes, b approximates a cylinder, and b a approximates a disk, or platelet. 

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